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| 1 | +--- |
| 2 | +layout: proof |
| 3 | +mathjax: true |
| 4 | + |
| 5 | +author: "Maja Pavlovic" |
| 6 | +affiliation: "Queen Mary University London" |
| 7 | +e_mail: "m.pavlovic@se22.qmul.ac.uk" |
| 8 | +date: 2022-10-02 09:02:00 |
| 9 | + |
| 10 | +title: "Variance of the log-normal distribution" |
| 11 | +chapter: "Probability Distributions" |
| 12 | +section: "Univariate continuous distributions" |
| 13 | +topic: "Log-normal distribution" |
| 14 | +theorem: "Variance" |
| 15 | + |
| 16 | +sources: |
| 17 | + - authors: "Taboga, Marco" |
| 18 | + year: 2022 |
| 19 | + title: "Log-normal distribution" |
| 20 | + in: "Lectures on probability theory and mathematical statistics" |
| 21 | + pages: "retrieved on 2022-10-01" |
| 22 | + url: "https://www.statlect.com/probability-distributions/log-normal-distribution" |
| 23 | + - authors: "Wikipedia" |
| 24 | + year: 2022 |
| 25 | + title: "Variance" |
| 26 | + in: "Wikipedia, the free encyclopedia" |
| 27 | + pages: "retrieved on 2022-10-01" |
| 28 | + url: "https://en.wikipedia.org/wiki/Variance#Definition" |
| 29 | + |
| 30 | +proof_id: "P355" |
| 31 | +shortcut: "lognorm-var" |
| 32 | +username: "majapavlo" |
| 33 | +--- |
| 34 | + |
| 35 | + |
| 36 | +**Theorem:** Let $X$ be a [random variable](/D/rvar) following a [log-normal distribution](/D/lognorm): |
| 37 | + |
| 38 | +$$ |
| 39 | +X \sim \ln \mathcal{N}(\mu, \sigma^2) . |
| 40 | +$$ |
| 41 | + |
| 42 | +Then, the [variance](/D/var) of $X$ is |
| 43 | + |
| 44 | +$$ \label{eq:lognorm-var} |
| 45 | +\mathrm{Var}(X) = \exp \left(2\mu +2\sigma^2\right) - \exp \left(2\mu + \sigma^2\right) \; . |
| 46 | +$$ |
| 47 | + |
| 48 | + |
| 49 | +**Proof:** |
| 50 | +[Variance](/D/var) is defined as: |
| 51 | + |
| 52 | +$$ \label{eq:var} |
| 53 | +\mathrm{Var}(X) = \mathrm{E}\left[ (X-\mathrm{E}(X))^2 \right] |
| 54 | +$$ |
| 55 | + |
| 56 | +which, [ partitioned into expected values](/P/var-mean) reads: |
| 57 | + |
| 58 | +$$ \label{eq:var2} |
| 59 | +\mathrm{Var}(X) = \mathrm{E}\left[ X^2 \right] - \mathrm{E}\left[ X \right]^2 |
| 60 | +$$ |
| 61 | + |
| 62 | +The [expected value of the log-normal distribution](/P/lognorm-mean) is: |
| 63 | + |
| 64 | +$$ \label{eq:lognorm-mean-ref} |
| 65 | +\mathrm{E}[X] = \exp \left( \mu + \frac{1}{2} \sigma^2 \right) |
| 66 | +$$ |
| 67 | + |
| 68 | +The second moment $\mathrm{E}[X^2]$ can be derived as follows: |
| 69 | + |
| 70 | +$$ \label{eq:second-moment} |
| 71 | +\begin{split} |
| 72 | +\mathrm{E} [X^2] = \int_{- \infty}^{+\infty} x^2 \cdot f_\mathrm{X}(x) \, \mathrm{d}x \\ |
| 73 | +&= \int_{0}^{+\infty} x^2 \cdot \frac{1}{x\sqrt{2 \pi \sigma^2} } \cdot \exp \left[ -\frac{1}{2} \frac{\left(\ln x-\mu\right)^2}{\sigma^2} \right] \mathrm{d}x \\ |
| 74 | +&= \frac{1}{\sqrt{2 \pi \sigma^2} } \int_{0}^{+\infty} x \cdot \exp \left[ -\frac{1}{2} \frac{\left(\ln x-\mu\right)^2}{\sigma^2} \right]\mathrm{d}x |
| 75 | +\end{split} |
| 76 | +$$ |
| 77 | + |
| 78 | +Substituting $z = \frac{\ln x -\mu}{\sigma}$, i.e. $x = \exp \left( \mu + \sigma z \right )$, we have: |
| 79 | + |
| 80 | +$$ \label{eq:second-moment-s2} |
| 81 | +\begin{split} |
| 82 | +\mathrm{E} [X^2] = \frac{1}{\sqrt{2 \pi \sigma^2} } \int_{(-\infty -\mu )/ (\sigma)}^{(\ln x -\mu )/ (\sigma)} \exp \left( \mu +\sigma z \right) \exp \left( -\frac{1}{2} z^2 \right) \mathrm{d} \left[ \exp \left( \mu +\sigma z \right) \right] \\ |
| 83 | +&= \frac{1}{\sqrt{2 \pi \sigma^2} } \int_{-\infty}^{+\infty} \exp \left( -\frac{1}{2} z^2 \right) \sigma \exp \left( 2\mu + 2 \sigma z \right) \mathrm{d}z \\ |
| 84 | +&= \frac{1}{\sqrt{2 \pi} } \int_{-\infty}^{+\infty} \exp \left[ -\frac{1}{2} \left( z^2 - 4 \sigma z - 4 \mu \right) \right] \mathrm{d}z \\ |
| 85 | +\end{split} |
| 86 | +$$ |
| 87 | + |
| 88 | +Now multiplying by $\exp \left( 2 \sigma^2 \right)$ and $\exp \left(- 2 \sigma^2 \right)$, this becomes: |
| 89 | + |
| 90 | +$$ \label{eq:second-moment-s3} |
| 91 | +\begin{split} |
| 92 | +\mathrm{E} [X^2] = \frac{1}{\sqrt{2 \pi} } \int_{-\infty}^{+\infty} \exp \left[ -\frac{1}{2} \left( z^2 - 4 \sigma z + 4 \sigma^2 -4 \sigma^2 - 4 \mu \right) \right] \mathrm{d}z \\ |
| 93 | +&= \frac{1}{\sqrt{2 \pi} } \int_{-\infty}^{+\infty} \exp \left[ -\frac{1}{2} \left( z^2 - 4\sigma z + 4\sigma^2 \right) \right] \exp \left( 2 \sigma^2 +2 \mu \right) \mathrm{d}z \\ |
| 94 | +&= \exp \left( 2 \sigma^2 +2 \mu \right) \int_{-\infty}^{+\infty} \frac{1}{\sqrt{2 \pi} } \exp \left[ -\frac{1}{2} \left( z - 2 \sigma \right)^2 \right] \mathrm{d}z \\ |
| 95 | +\end{split} |
| 96 | +$$ |
| 97 | + |
| 98 | +The [probability density function of a normal distribution](/P/norm-pdf) is |
| 99 | + |
| 100 | +$$ |
| 101 | +f_X(x) = \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2 \right] |
| 102 | +$$ |
| 103 | + |
| 104 | +with $\mu = 2 \sigma$ and unit variance this reads: |
| 105 | + |
| 106 | +$$ |
| 107 | += \frac{1}{\sqrt{2 \pi}} \cdot \exp \left[ -\frac{1}{2} \left({x - 2 \sigma} \right)^2 \right] |
| 108 | +$$ |
| 109 | + |
| 110 | +Using the definition of the [probability density function](/D/pdf) |
| 111 | + |
| 112 | +$$ |
| 113 | +\int_{-\infty}^{+\infty} \frac{1}{\sqrt{2 \pi}} \cdot \exp \left[ -\frac{1}{2} \left({x - 2 \sigma} \right)^2 \right] \mathrm{d}x = 1 |
| 114 | +$$ |
| 115 | + |
| 116 | +and applying \eqref{eq:def-pdf} to \eqref{second-moment-s3}, we have: |
| 117 | + |
| 118 | +$$ \label{eq:second-moment-4} |
| 119 | +\mathrm{E}[X]^2 = \exp \left( 2 \sigma^2 +2 \mu \right) \; . |
| 120 | +$$ |
| 121 | + |
| 122 | +Applying \eqref{eq:second-moment-4} and \eqref{eq:lognorm-mean-ref} to \eqref{eq:var2}, we have: |
| 123 | + |
| 124 | +$$ \label{eq:lognorm-var-2} |
| 125 | +\begin{split} |
| 126 | +\mathrm{Var}(X) = \mathrm{E}\left[ X^2 \right] - \mathrm{E}\left[ X \right]^2 \\ |
| 127 | +&= \exp \left(2\sigma^2 + 2\mu \right) - \left[ \exp \left(\mu + \frac{1}{2} \sigma^2 \right) \right]^2 \\ |
| 128 | +&= \exp \left(2\sigma^2 + 2\mu \right) - \exp \left(2\mu + \sigma^2\right) \; . |
| 129 | +\end{split} |
| 130 | +$$ |
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